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Exponents and Roots

  • Exponents are used to denote the repeated the multiplication of a number by itself; for e.g. = (3)(3)(3)(3) = 81 . In the expression 34, 3 is called the base, 4 is called the exponent, and we read the expression as "3 to the fourth power." When the exponent is 2, we call the process squaring. Thus 7 squared is 49, 72 = (7)(7) = 49.
  • A negative number raised to an even power is always positive, and a negative number raised to an odd power is always negative. Note that without the parentheses, the expression -32 means "the negative of '3 squared'"; that is, the exponent is applied before the negative sign. So (-3)2 = 9, but -32 = -9.
  • Exponents can also be negative or zero; such exponents are defined as follows.
  • For all nonzero numbers a, a0 = 1. The expression 00is undefined.
  • For all nonzero numbers a, , etc. Note that .
  • A square root of a nonnegative number  is a number  such that . For e.g. 4 is a square root of 16 because . Another square root of 16 is -4, since . All positive numbers have two square roots, one positive and one negative. The only square root of 0 is 0. The symbol  is used to denote the nonnegative square root of the nonnegative number . Therefore, and . Square roots of negative numbers are not defined in the real number system.
  • Here are some important rules regarding operations with square roots, where  and

 

Rule

Examples

 

 

 

   

  • A square root is a root of order 2. Higher- order roots of a positive number  are defined similarly. For orders 3 and 4, the cube root  and fourth root  represent numbers such that when they are raised to the powers 3 and 4 respectively, the result is  These roots obey rules similar to those above (but with the exponent 2 replaced by 3 or 4 in the first two rules).
  • For odd-order roots, there is exactly one root for every number , even when  is negative.
  • For even-order roots, there are exactly two roots for every positive number  and  no roots for any negative number
  • For e.g. 8 has exactly one cube root, but 8 has two fourth roots   and ; and -8 has exactly one cube root, , but -8 has no fourth root, since it is negative.
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